Finite Math Examples

$\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.5

Add $1$ and $1$ .

$\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{\sqrt{2} \sqrt{5}}{5^{1}} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 2.6.5

Evaluate the exponent.

$\frac{\sqrt{2} \sqrt{5}}{5} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 3

Simplify the numerator.

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Step 3.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 5}}{5} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 3.2

Multiply $2$ by $5$ .

$\frac{\sqrt{10}}{5} \cdot \frac{\sqrt{10}}{\sqrt{5}}$

Step 4

Combine $\sqrt{10}$ and $\sqrt{5}$ into a single radical.

$\frac{\sqrt{10}}{5} \cdot \sqrt{\frac{10}{5}}$

Step 5

Divide $10$ by $5$ .

$\frac{\sqrt{10}}{5} \cdot \sqrt{2}$

Step 6

Multiply $\frac{\sqrt{10}}{5} \sqrt{2}$ .

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Step 6.1

Combine $\frac{\sqrt{10}}{5}$ and $\sqrt{2}$ .

$\frac{\sqrt{10} \sqrt{2}}{5}$

Step 6.2

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 2}}{5}$

Step 6.3

Multiply $10$ by $2$ .

$\frac{\sqrt{20}}{5}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.1.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{5}$

Step 7.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 5}}{5}$

Step 7.2

Pull terms out from under the radical.

$\frac{2 \sqrt{5}}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{5}}{5}$

Decimal Form:

$0.89442719 \dots$